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R0017-09042 · q-20261013-advance-20261018-r05-modular-congruence-0042
Fix positive modulus n=9, natural a=49, bound=41. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).
G001 — Fix positive modulus n=9, natural a=49, bound=41. Give (n, a, bound, a mod n, every x∈[0,bound] with x mod n=a mod n).Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
fixed natural modulus candidate=9
After
positive modulus=Yes
Conditions: none
Congruence compares remainders at one positive modulus,not numerical equality.Multiplicative preservation does not permit unconditional cancellation of a common factor.
Rule: fixed_positive_modulus_scope · Uses: G001
Before
representative remainder:natural dividend=49;positive divisor=9
After
49=5·9+4
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S001
Before
window bound minus residue := sub(41, 4)
After
window bound minus residue = 37
Conditions: none
Calculate exactly in the specified source-operand order without approximation or substituting a different candidate.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
last progression index:natural dividend=37;positive divisor=9
After
37=4·9+1
Conditions: none
With a positive divisor show the complete quotient/remainder equality and0≤r<divisor.Modulo zero is not evaluated.
Rule: positive_euclidean_division · Uses: G001, S003
Before
residue=4;modulus=9;0<=k<=4
After
all class members=[4, 13, 22, 31, 40]
Conditions: none
Euclidean uniqueness gives x≡a exactly when x=r+k*n.Natural x and0≤r<n force k≥0;the bound gives k≤floor((bound−r)/n).Retain the full inclusive progression,not a sample.
Rule: complete_residue_progression · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Recompute window bound minus residue with exact rational arithmetic, not rounded decimals.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(9, 49, 41, 4, [4, 13, 22, 31, 40])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: ce562fc3e9272a7dea7787fb7a12b4524265692ce53539540cd55c5a65f99878candidate record SHA-256: 102b1ab492c312d834fc9bcce219ab8c9e1f1cb255c7cffa66d4ce769185b55fsemantic review receipt SHA-256: abad28564830bb9b6ef069ca5a1aa128ce7b1a67a5cb8fe6d868a06db73bcfb8question-first witness SHA-256: 43126988479968837abe086263dac1b24c37463ad80e00d2e280a7b8e9036a04Lean: NOT_RUN · human review: NOT_PERFORMED